1 / 19

Body size distribution of European Collembola

Lecture 9 Moments of distributions. Body size distribution of European Collembola. Body size distribution of European Collembola. Modus. The histogram of raw data. Three Collembolan weight classes. What is the average body weight ? . Sample mean. Population mean.

nan
Télécharger la présentation

Body size distribution of European Collembola

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Lecture 9 Moments of distributions Body sizedistribution of EuropeanCollembola

  2. Body sizedistribution of EuropeanCollembola Modus The histogram of raw data

  3. ThreeCollembolanweightclasses Whatistheaverage body weight? Sample mean Population mean Weighedmean

  4. Weighed mean Discrete distributions TheaverageEuropeanspringtailhas a body weight of e-1.476 = 023 mg. Most oftenencountedis a weightaround e-1.23 = 029 mg. Continuous distributions

  5. Whydid we use log transformedvalues? Linear data Log transformed data Thedistributionisskewed

  6. lbscaledweightclasses TheaverageEuropeanspringtailhas a body weight of e-1.476 = 023 mg. Geometricmean In thecase of exponentiallydistributed data we have to usethegeometricmean. To make thingseasier we first log-transformour data.

  7. How to usegeometricmeans A tropical forest is logged during three years: first year 0.1%, second year 1% and third year 10% of area. Hence the total decrease in forest area is 11% of area has been logged during three year. What is the mean logging rate per year? Arithmetic mean Geometric mean In multiplicativeprocesses we shouldusethegeometricmean.

  8. Degrees of freedom Variance Continuousdistributions Mean 1 SD Standard deviation The standard deviationis a measure of thewidth of thestatisticaldistributionthathasthe sam dimension as themean.

  9. The standard deviation as a measure of errors ± 1 standard deviationisthe most oftenusedestimator of error. Theprobablitythatthetruemeaniswithin± 1 standard deviationisapproximately 68%. Theprobablitythatthetruemeaniswithin± 2 standard deviationsisapproximately 95%. The precision of derivedmetricsshouldalwaysmatchthe precision of theraw data ± 1 standard deviation

  10. Standard deviation and standard error The standard deviationisconstantirrespective of samplesize. The precision of theestimate of themeanshouldincreasewithsamplesize n. The standard erroris a measure of precision.

  11. Central moments [E(x)]2 E(x2) Mathematicalexpectation First central moment First moment of central tendency Thevarianceisthedifferencebetweenthemean of thesquaredvalues and thesquaredmean k-th central moment

  12. Frequency distributions of resource use or wealth in a population can be described by a power law (the famous Pareto-Zipf law) with exponents that often have values around -5/2. Whatare the mean and thevarianceof such a power function distribution? Discretedistribution Most peopleareinthelowestincomeclass and theaverageishalfbetweenthe first and thesecond.

  13. Continuousapproximation Notethatthey-axisisat log scale. Upper bound of ten wouldonlycoverhalf of thecolumn Theestimate of a isimprecise

  14. The Arrhenius probability model assumesthe same probability of an eventirrespective of the time thatelapsedfromthestarting. Whatarethe mean and thevarianceof such a distribution? Cumulativedensityfunction

  15. Third central moment Skewness g>0 g<0 g=0 Leftskeweddistribution Rightskeweddistribution Symmetricdistribution d>0 d=0 Kurtosis

  16. How to getthe modus? We needthemaximum of thepdf A probabilitydistributionif Mode Mean Arithmeticmean

  17. Body volumes are estimated from measures of height*length*width. Assume you estimated the thorax volume of insects and usedthisvolume to infer the body weight. How to gettheparameters a and z?

  18. Body weightsareestimated from speciesweightsagainstthoraxvolume. The body weight of a newspeciesisestimatedfromtheregressionfunction Height, length and width could be measured with an accuracy of ± 2%. Independent measurements Standarddeviation is a measure of accuracy (error) Theerror of thethoraxestimateis 3.5%.

  19. Home work and literature • Refresh: • Arithmetic, geometric, harmonicmean • Cauchyinequality • Statisticaldistribution • Probabilitydistribution • Moments of distributions • Error law of Gauß • Bootstrap • Prepare to thenextlecture: • Bionomialdistribution • Mean and variance of thebinomialdistribution • Poisson distribution • Mean and variance of the Poisson distribution • Moments of distributions • DNA mutations • Transitionmatrix Literature: Łomnicki: Statystyka dla biologów Binomialdistribution: http://www.stat.yale.edu/Courses/1997-98/101/binom.htm Poisson dstribution: http://en.wikipedia.org/wiki/Poisson_distribution

More Related